On nonlocally coupled complex Ginzburg-Landau equation
Dan Tanaka and Yoshiki Kuramoto1
Department of Physics, Graduate School of Sciences, Kyoto University, Kyoto 606-8502, Japan
arXiv:nlin/0304017v1 [nlin.PS] 11 Apr 2003
A Ginzburg-Landau type equation with nonlocal coupling is derived systematically as a reduced form of a universal class of reaction-di?usion systems near the Hopf bifurcation point and in the presence of another small parameter. The reaction-di?usion systems to be reduced are such that the chemical components constituting local oscillators are non-di?usive or hardly di?usive, so that the oscillators are almost uncoupled, while there is an extra di?usive component which introduces e?ective nonlocal coupling over the oscillators. Linear stability analysis of the reduced equation about the uniform oscillation is also carried out. This revealed that new types of instability which can never arise in the ordinary complex Ginzburg-Landau equation are possible, and their physical implication is brie?y discussed.
PACS numbers: 05.45.-a,47.54.+r,82.40.-g
Oscillatory reaction-di?usion systems are generally reduced to the complex GinzburgLandau (CGL) equation by means of the so-called center-manifold reduction when the local oscillators are close to their supercritical Hopf bifurcation point. This fact led to a now widely accepted view that, without resorting to individual systems, one may concentrate on CGL if one wishes to gain a qualitative understanding of some universal dyanmical features shared commonly by a broad class of oscillatory reaction-di?usion systems. This view in fact underlies a vast amount of work in the past (see [2, 3] for review) devoted to CGL. However, due to its local nature of coupling, CGL may fail to capture some important aspects of the dynamics characteristic to a certain class of oscillatory reaction-di?usion systems. In fact, it was argued recently that a situation may arise where the oscillator coupling becomes e?ectively nonlocal, and as a consequence the system exhibits such peculiar dynamics as can never be seen in CGL [4, 5]. This suggests that there remains a yet unexplored area of reaction-di?usion systems where nonlocal e?ects on the pattern dynamics must seriously be considered. In the present paper, we are concerned with how this area is still accessible within the framework of the center-manifold reduction. It will turn out that this can actually be achieved by a slight extension of the conventional reduction scheme. E?ective nonlocality in coupling may become relevant when the reaction-di?usion system involves three or more chemical components. Suppose that the system of concern is such that the chemical components constituting the local oscillators are free of di?usion, i.e., the oscillators remain uncoupled, while the system involves an extra di?usive component which, for its di?usive nature, plays the role of a coupling agent. By eliminating mathematically this di?usive component, the system becomes a ?eld of nonlocally coupled oscillators possibly involving delay also. The main goal of the present paper is to achieve a reduction of such reaction-di?usion systems to a universal equation of the Ginzburg-Landau type without missing the e?ects of nonlocality. It may seem that the center-manifold idea would not work for our purpose because CGL is believed to be quite a general result of the center-manifold reduction. Still there seems to be a way out within the same framework if we slightly extend the conventional method of reduction. Note ?rst that under usual conditions the disappearance of the e?ects of nonlocality near the bifurcation point comes from that the characteristic wavelength lp of 2
the pattern becomes far longer than the e?ective coupling radius. If so, it may happen that nonlocality persists even close to the bifurcation point when the system involves a certain parameter whose suitable tuning as the bifurcation point is approached keeps lp comparable with the e?ective radius of coupling. Similar physical idea lies behind the multiple bifurcation theory which aims to capture such complex dynamics as is absent in the vicinity of a simple bifurcation point. Section II starts with introducing a universal class of oscillatory reaction-di?usion systems involving three or more chemical components. After brie?y discussing its physical relevance, we proceed to its reduction near the Hopf bifurcation. If a certain parameter associated with the strength of e?ective nonlocal coupling is as small as the bifurcation parameter, the reduced equation turns out to take the form of a nonlocally coupled complex GinzburgLandau equation. Analytic formulae for some coe?cients of this equation are given, which would be of great help in inferring possible ranges of the parameters in the original system where nonlocality-dominated pattern dynamics is expected. Regarding the derivation of the reduced equation, our primary concern is the case when direct oscillator coupling is absent, but the e?ects of weak di?usive (i.e. direct) coupling will also be considered. In Section III, linear stability analysis of our nonlocal CGL is carried out about the uniform oscillation. The resulting eigenvalue spectra, especially those of the phase branch, can be qualitatively di?erent from those of the standard CGL, whose physical implication is discussed. A short summary will be given in the ?nal section.
A UNIVERSAL CLASS OF REACTION-DIFFUSION SYSTEMS AND THEIR
The reaction-di?usion model of our concern was previously proposed by one of the present authors and is given by the general form ?t X = f (X) + kg(S), τ ?t S = ?S + D?2 S + h(X). (1) (2)
Here the n-dimensional real vector ?eld f represents a local limit-cycle oscillator with dynamical variable X, so that the ?rst equation with k = 0 represents a ?eld of continuously distributed oscillators without mutual coupling. The full system involves an additional chem3
ical component with concentration S whose dynamics is governed by the second equation. This component simply di?uses and decays at a constant rate, while it is produced locally as represented by the term h(X), the production rate depending on the local value of X. The dynamics of the local oscillators is in?uenced in return by the local concentration of S, and this e?ect is represented by the term kg(S). For the sake of convenience, we inserted in Eq. (2) parameter τ to indicate explicitly the time constant of S, anticipating a limiting case in which τ is vanishingly small. Note that there is no direct coupling among the local oscillators, while their indirect coupling is provided by the di?usive chemical represented by S. In later discussions, we will generalize the above model by assuming g and h to depend both on X and S , and also by including a small di?usion term i.e. weak direct coupling in Eq. (1). Throughout the present paper, the spatial extension of the system is supposed su?ciently large. Let the spatially uniform steady state of our system be given by (X, S) = (0, 0), or equivalently, we measure X and S always from their equilibrium values. Thus, we clearly have h(0) = 0. It is also convenient to arrange f and g so that the equalities f (0) = g(0) = 0 may be satis?ed. If we like, S may be eliminated from the system by solving Eq. (2), which can be done explicitly because the equation is linear. If the spatial dimension is d, the solution of Eq. (2) is given by S(r, t) = (2π)?d dq exp(iqr)
′ t ?∞
dt′ τ (3)
exp ?(1 + Dq 2 )
t?t hq (t′ ), τ
where hq (t′ ) is the spatial Fourer transform of h(X(r, t′ )). Equation (1) with S given by Eq. (3) constitutes a ?eld of nonlocally coupled oscillators. Note that the nonlocality appears in time as well as in space, and the characteristic scales in time and space associated with the nonlocality are given by τ and D 1/2 , respectively. This fact will be used in the discussion below. Suppose that f involves a parameter ? such that if ? < 0 each local system given by Eqs. (1) and (2) with D = 0 has a stable ?xed point (X, S) = (0, 0) while this becomes oscillatory unstable for ? ≥ 0; namely, at ? = 0 a pair of complex conjugate eigenvalues of the Jacobian associated with each local system about the ?xed point cross the imaginary axis of the complex plane, while the other eigenvalues all remain in the left half plane. It is a well known fact that the center-manifold reduction can be applied to reaction-di?usion 4
systems near the Hopf bifurcation point of the local oscillators. This leads generally to the complex Ginzburg-Landau (CGL) equation ?t A = ?σA ? β|A|2A + α?2 A, (4)
where the amplitude A and the parameters σ, α and β are generally complex. The small parameter ? has been retained in Eq. (4) so that A, t and r scale like |?|1/2 , |?|?3/2 and |?|?1/2 , respectively. It is clear that the above reduced form of reaction-di?usion systems remains valid also for our particular system given by Eqs. (1) and (2). Since CGL is a di?usively coupled (i.e. locally coupled) system, it may seem that e?ective nonlocality in coupling characteristic to our system disappears completely near the bifurcation point. The reason for the disappearance of nonlocality is clear. This is because the characteristic wavelength lp of the ?eld X the e?ective coupling radius given by D 1/2 comes to fall well within this scale, which gives becomes longer and longer as the bifurcation point is approached like lp ? |?|?1/2 so that
nothing but the de?nition of local coupling. In what follows, we will be concerned with a special situation in which spatial (and possibly temporal) nonlocality can survive even close to the bifurcation point, so that the reduced equation involves nonlocal coupling rather than di?usive coupling. The same result was used already in earlier works[4, 5] for the particular case of vanishingly small τ without showing how the reduction can actually be achieved. We will develop below the reduction procedure explicitly including the case of ?nite τ . Consider our system given by the form of Eqs. (1) and (3) for which Eq. (4) gives the right reduced form near ? = 0 provided there is no small parameter other than ?. It is clear that the di?usion term in Eq. (4) is the reduced form of the coupling term kg(S) in Eq. (1). This implies that |α| = O(|k|). Disappearance of spatial nonlocality is consistent with the fact that the characteristic wavelength lp estimated from the dimensional argument for Eq. (4), which con?rms lp = O(|?/k|?1/2), is far larger than the e?ective coupling radius given by D 1/2 provided k remains an ordinary magnitude. This consistency is apparently broken if k becomes as small as ? by which lp becomes independent of ? and hence can be comparable with the coupling radius. Putting it di?erently, spatial nonlocality should remain relevant near the bifurcation point provided the strength of coupling between the local oscillators and the di?usive component becomes so weak as to satisfy k ? O(|?|). 5 (5)
Thus, what we do next is to ?nd a reduced equation valid near the doubly singular point (?, k) = (0, 0). Before proceeding to this issue, however, we make a remark on temporal nonlocality. Temporal nonlocality, which is characterized by the time constant τ , may be relevant to the dynamics even near the bifurcation point. However, this e?ect would never appear as a non-Marko?an form of the reduced equation for the amplitude A because the time scale associated with the variation of A is much longer than τ . Still, τ could be comparable with the period of the basic oscillation. Then, as we see below, the e?ect of delay may appear in the reduced equation through the change in the space-dependence of the coupling function. Speci?cally, when τ is ?nite, the decay of the coupling function with distance becomes oscillatory rather than monotone. Suppose ?rst that k = 0. Then the reduced form of the ordinary equation (1) is given by Eq. (4) with α = 0. Although its derivation is routine, we now recapitulate it for the purpose of explaining soon later how easily the reduced equation can be generalized when a small coupling term is introduced. Let the Taylor expansion of f (X) in terms of X be written as f (X) = LX + MXX + NXXX + · · · . (6)
Regarding the ?-dependence of the coe?cients appearing in the above expansion, we need to consider it only for the Jacobian L to the ?rst order in ? like L = L0 + ?L1 ; higher order corrections to L as well as ?-dependence of M, N etc. are irrelevant to the reduced equation to the leading order. Let the pure imaginary eigenvalues at ? = 0 be ±iω0 , and ? the corresponding right eigenvectors and its complex conjugate are written as U and U , respectively. The corresponding left eigenvectors and its complex conjugate are denoted by ? ?? ??? ?? U ? and U , respectively. These eigenvectors satisfy U ? U = U U = 0 and U ? U = U U = ? 1. If ? is nonvanishing but small, the eigenvalues change to ±iω0 + ?λ± , where λ+ = λ? . To the lowest order in ?, the original ?eld X and the complex amplitude A are mutually related via X(t) = eiω0 t U A(t) + c.c.. (7)
˙ Thus, in this approximation we have ?t A = exp(?iω0 t)U ? (X ? L0 X). This means further that the right-hand side of Eq. (4) with α = 0 is identical with the reduced form of exp(?iω0 t)U ? (f (X) ? L0 X), or ?σA ? β|A|2A ? e?iω0 t U ? f (X) ? L0 X 6
= e?iω0 t U ? (?L1 X + MXX + NXXX + · · ·).
The standard analysis determines the coe?cients σ and β in terms of some parameters of the equations before reduction. It is clear that the linear coe?cient σ is given by σ = U ? L1 U = λ+ . (9)
For obtaining β, the lowest order expression for X given by Eq. (7) is insu?cient, and we have to use a more precise formula including the next order term: X(t) = eiω0 t U A(t) + c.c. ? + e2iω0 t V + A2 + e?2iω0 t V ? A2 + V 0 |A|2 , where ? V + = V ? = ?(L0 ? 2iω0 )?1 M U U , ? V 0 = ?2L?1 M U U . 0 Using these quantities, β is given by ? ? β = ?2U ? MU V 0 ? 2U ? M U V + ? 3U ? NU U U . (13) (11) (12) (10)
Suppose that the vector ?eld f is modi?ed slightly to f +p. It is clear that the corresponding reduced equation must also be modi?ed slightly with an additive term exp(?iω0 t)U ? p. The speci?c form of p of our concern is the small coupling term kg(S) in Eq. (1) with S given by Eq. (3). The original variable X and the reduced one A must now be regarded as depending on space as well as on time. Thus, our problem is to ?nd a reduced form of the quantity k exp(?iω0 t)U ? g(S) ≡ k? p ? bution to p. Noting that g(0) = 0, we may use a linear approximation g(S) ? g 0 S g0 ? (2π)d (14)
using Eq. (3). Since k is already small, we only need to consider the most dominant contri-
dt′ τ (15)
exp ?(1 + Dq 2 ) 7
t?t h0 X q (t′ ), τ
where g 0 = dg/dS|S=0 and h0 = dh(X)/dX|X =0 . Thus, using Eq. (7) with X and A supposed to depend also on r, we have ? p = η (2π)d dq exp(iqr)
dt′ τ (16)
t ? t′ exp ? (1 + Dq 2 ) ? iω0 (t ? t′ ) Aq (t′ ), τ where η = (U ? g 0 )(h0 U ).
Note that Eq. (16) ignores the contribution from the complex conjugate of A(t) which would ? give rise to a rapidly oscillating component of p. This is allowed because such component would be averaged out in the equation describing the slow evolution of A(t). Since the time-integral in Eq. (16) extends practically over the interval between t ? τ and t, the slowly varying amplitude Aq (t′ ) may safely be replaced with Aq (t). In this way, we obtain k? = kη ′ p where G(r) = and η′ = η . 1 + iω0 τ (20) 1 (2π)d dqeiqr 1 + iω0 τ , Dq 2 + 1 + iω0 τ (19) dr′ G(r ? r ′ )A(r ′ , t),
Note that G(r) satis?es the normalization condition drG(r) = 1. Thus the ?nal form of the reduced equation becomes ?t A = ?σA ? β|A|2 A +kη ′ dr′ G(r ? r ′ )A(r ′ , t) (22) (21)
which we call nonlocally coupled complex Ginzburg-Landau equation or simply nonlocal CGL. It is clear that the situation of interest is such that k = O(|?|) for which the coupling term in the reduced equation is balanced in magnitude with the other terms even if the characteristic wavelength is independent of ?. We assume that the bifurcation is supercritical, i.e., the real part of β is positive. 8
A few generalizations of the original system (Eqs. (1) and (2)) can be made. Firstly, g and h may depend both on X and S. Since the most dominant part of these quantities alone is relevant to the reduced equation, one may safely approximate g(X, S) and h(X, S) as g(X, S) = g(X, 0) + g(0, S) and h(X, S) = h(X, 0) + h(0, S), respectively. The resulting new term kg(X, 0) may slightly modify f (X), but the modi?ed f (X) may again be denoted by f (X). Furthermore, because S is small, h(0, S) is practically linear in S. Thus, this quantity simply modi?es the linear decay rate of S which can be normalized by a suitable rescaling of time. In this way, the ?nal result of reduction is unchanged except that g(S) and h(X) are replaced with g(0, S) and h(X, 0), respectively. As the second generalization, we may include in Eq. (1) a di?usion term like ? ?t X = f (X) + δ?2 X + kg(S), (23)
? where δ is a diagonal di?usion matrix with non-negative elements. In parallel with the above argument for obtaining a reduced form of the nonlocal coupling term, the reduced form of ? the quantity exp(?iω0 t)U ? δ?2 X will then be added to the right-hand side of Eq. (22). To the lowest order approximation, one may apply Eq. (7) for X, by which the above quantity becomes δ?2 A where δ is a complex number with positive real part, and is given by ? δ = U ? δU . Thus, Eq. (22) is modi?ed as ?t A = ?σA ? β|A|2A + δ?2 A +kη ′ dr ′ G(r ? r ′ )A(r ′ , t). (25) (24)
In the conventional reduction of reaction-di?usion systems, |δ| is assumed to be of ordinary magnitude, so that the di?usion term can be balanced with the linear and cubic terms in magnitude only if the characteristic wavelength of A scales like |?|?1/2 . However, the last property of A contradicts with the particular situation of our concern in which the characteristic wavelength remains independent of ?. Therefore, in what follows, we assume that |δ| as well as k is of the order of ?, by which all terms on the right-hand side of Eq. (25) are balanced with each other, and the coupling nonlocality represented by the last term can survive.
It is more convenient to write Eq. (25) in the form ?t A = ?σ ′ A ? β|A|2 A + δ?2 A + kη ′ dr′ G(r ? r ′ )(A(r ′ , t) ? A(r, t)), (26)
where σ ′ = σ + ??1 kη ′ . With this form, the coupling term can be approximated by a di?usion term when the characteristic wavelength of A(r, t) is su?ciently longer than the coupling radius. Note that σ ′ remains ordinary magnitude because k = O(?) by assumption. Hereafter, we assume that the system is supercritical or Reσ ′ > 0. (27)
An additional remark should be made on the functional form of the coupling function G in connection with the time scale τ of the di?usive component S. As implied by Eq. (16), ?nite τ generally gives rise to memory e?ects or temporal nonlocality after the variable S has been eliminated. However, as was noted before, the reduced equation is free from memory e?ects because the time scale of the slowly varying amplitude A is much longer than τ . We also noted that because τ may be comparable with the period 2π/ω0 of the fundamental oscillation, the e?ect of delay in coupling should be relevant to the reduced dynamics. Actually, as is clear from Eq. (19), the e?ect of ?nite τ appears in the coupling function G. For one-dimensional systems, in particular, the coupling function is simply expressed as 1 G(x) = (α+ + iα? )e?(α+ +iα? )|x| , 2 where α± = ±1 + 1 + θ2 2D √ (28)
and θ = ω0 τ . If τ is vanishing, G(x) decays exponentially with |x|, the decay length being delay in coupling is such that the complex amplitude A(x′ ) in the coupling term is multiplied
given by D 1/2 , whereas the decay becomes oscillatory when τ is ?nite. Thus the e?ect of
by a factor exp(?iα? |x ? x′ |), or equivalently, the phase of A(x′ , t), denoted by φ(x′ , t), is the spatial point x′ experienced by the oscillator at x through the delayed coupling cannot be its current value φ(x′ , t) but should be the value at some time t ? t0 in the past, because the phase information travels at a ?nite speed. If this speed is constant, and the oscillation 10
replaced with φ(x′ , t) ? α? |x ? x′ |. In the physical language, this means that the phase at
at x′ is nearly regular, then we have φ(x′ , t?t0 ) equal to φ(x′ , t) plus something proportional to the distance |x ? x′ |, justifying the above result. Coming back to general space dimensions, Eq. (26) with the coupling function given by Eq. (19) involves many parameters. However, some of the parameters can be eliminated by suitable transformations of some variables. Firstly, the imaginary part of the linear coe?cient ?σ ′ vanishes through the transformation A → A exp[(i?Imσ ′ )t]. Secondly, one way, we may write Eq. (26) as may rescale A, t and r in such a way that Reβ, ?Reσ ′ and D may all become unity. In this
?t A = A ? (1 + ic2 )|A|2 A + (δ1 + iδ2 )?2 A +K(1+ic1 ) dr ′ G(r ? r ′ )(A(r ′ , t) ? A(r, t)), where the coupling function G is de?ned as an integral form given by Eq. (19), or G(r) = 1 dqeiqr Gq , (2π)d 1 + iθ . Gq ≡ 2 q + 1 + iθ (31) (32) (30)
The reduced equation now involves six independent parameters c1 , c2 , K, θ, δ1 and δ2 all of which are independent of smallness parameters ?, k and δ. Note that the coupling coe?cient in Eq. (30) is related to some original parameters through K(1 + ic1 ) = or Reσ (34) Reσ ′ so that, by combining the last equation with the inequality (27) and the original assumption K =1? Reσ > 0, we have a restrictive condition K < 1. (35) kη ′ kη ′ = , ?Reσ ′ ?Reσ + kReη ′ (33)
EIGENVALUE SPECTRUM ABOUT THE UNIFORM OSCILLATION
It is clear that Eq. (30) admits a family of plane wave solutions Ak (r, t) = R exp[i(kr ? ?t)] the stability of which is extremely important to the understanding of the pattern dynamics of our system. In the present paper we will focus on the stability of the uniform 11
oscillation A0 (t) for which R = 1, ω0 = ?c2 . We now put A(r, t) = 1 + ?(r, t) A0 (r, t), (38) (36) (37)
and linearize Eq. (30) in ?(r, t). The linearized equation can be solved in terms of the Fourier components of ?(r, t), denoted by ?q (t). Assuming its time-dependence in the form ?q (t) ∝ exp(λt), we ?nd the eigenvalue equation λ2 + u(q)λ + v(q) = 0, where u(q) = ?2Reγ(q), v(q) = |γ(q)|2 ? |γ(0)|2, with γ(q) = ?(1 + ic2 ) ? (δ1 + iδ2 )q 2 +K(1 + ic1 )(Gq ? G0 ). (42) (40) (41) (39)
In what follows, we shall ?rst concentrate on the case without di?usive coupling, i.e. δ1 = δ2 = 0; the e?ects of nonvanishing but small δ1 will be touched upon later. Let the solutions of Eq. (39) be denoted by λ+ and λ? with Reλ+ ≥ Reλ? . The uniform oscillation is stable if and only if Reλ+ < 0 holds for all q. This holds only when u(q) > 0 and v(q) > 0 for all q. Equivalently, if one of these inequalities becomes violated for a certain q, then the uniform oscillation loses stability. This implies that the types of instability could be classi?ed in terms of the signs of u(q) and v(q). Simple calculation shows that u(q) and v(q) can be expressed in the following form where we use the notation Q ≡ q 2 . u(q) = ξ(Q)ξ0 (Q), v(q) = ζ(Q)ζ0(Q)Q. (43) (44)
Here ξ0 (Q) and ζ0 (Q) are non-negative functions of Q, and ξ(Q) ≡ a2 Q2 + a1 Q + a0 , ζ(Q) ≡ b1 Q + b0 , a0 = 1 + θ2 , a1 = 2 + K(1 + c1 θ), a2 = 1 + K, b0 = 2K(1 + c1 c2 + (c1 ? c2 )θ), b1 = K(2(1 + c1 c2 ) + K(1 + c2 )). 1 (45) (46) (47) (48) (49) (50) (51)
Figure 1 shows the a1 -a2 plane divided into three domains (labeled by A, B and C) corresponding to qualitatively di?erent forms of u(q). Note that the a1 -a2 plane covers all possible types of u(q) because a0 is positive de?nite. Similar picture for v(q) in the b0 -b1 plane is displayed in Fig. 2 where the whole space is divided into four domains (labeled by A′ to D′ ). Since the parameters a1 , a2 , b0 and b1 as functions of four independent parameters K, θ, c1 and c2 can be changed independently, every combination between two members, one from the group (A,B,C) and the other from (A′ ,B′ ,C′ ,D′ ), is possible. Uniform oscillation is unstable for all these combinations except for (A,A′ ). Loss of stability occurs as we move across the line a2 = 0 or a2 ? 4a0 a2 = 0 in Fig. 1, for which the instability is oscillatory, 1 or otherwise b0 = 0 or b1 = 0 in Fig. 2, for which the instability is non-oscillatory. The critical wavenumber for which Reλ < 0 becomes ?rst violated equals zero on the line b0 = 0, ?nite on a2 ? 4a0 a2 = 0 and in?nite on a2 = 0 and b1 = 0. The last type of instability, i.e. 1 the instability which starts at in?nite q, was called short-wavelength bifurcation by Heagy et al.. If |K| is not too large, u is non-negative, so that the instability is only through v becoming negative and hence it is non-oscillatory. The corresponding critical lines b0 = 0 and b1 = 0 in Fig. 2 can now be translated into critical relations between K and Kc1 under ?xed c2 and θ. In this way, we have four types of eigenvalue spectrum shown schematically in Fig. 3 where lines L1 and L2 correspond to b0 = 0 and b1 = 0, respectively. The ?gure shows the spectra only for phase-like ?uctuations, because in each case the amplitude branch which is separated from the phase branch remains negative for all q so that the amplitudelike ?uctuations are irrelevant to stability. Separation between the phase and amplitude 13
branches also implies that the eigenvalues associated with these branches are all real. In contrast, if these branches merge, then the eigenvalues of the two branches for given q would form a complex conjugate pair. Note that in each type of spectrum shown in the ?gure the eigenvalue saturates to a constant as the wavenumber tends to in?nity, which is characteristic to nonlonally coupled systems. The characteristic wavenumber about which the eigenvalue starts to saturate equals the inverse of the coupling radius. Physically, this re?ects the fact that the dynamics of ?uctuations whose wavelengths are much shorter than the coupling radius is practically the same as the oscillators’ individual dynamics, and hence it is insensitive to the wavenumber. Note also that the type U2 spectrum is possible only when θ is nonvanishing or, equivalently, when the nonlocal coupling before reduction involves delay. For larger |K|, the phase and amplitude branches can merge, so that oscillatory instability becomes possible. In particular, the short-wavelength type instability i.e. the instability initiated by ?uctuations with in?nite wavenumber can occur across the line a2 = 0 or K = ?1. This type actually appears in the stability diagram in Fig. 4 which is a global extension of the diagram in Fig. 3. The values of the other parameters are the same as in Fig. 3, i.e., c2 = 1.0 and θ = 6.0. Oscillatory instability initiated by ?uctuations with ?nite wavenumber, which occurs when crossing the line a2 ? 4a0 a2 = 0 in Fig. 1, does not appear 1 in Fig. 4, but may appear when di?erent values of c2 and θ are chosen. For instance, the stability diagram for the case of c2 = ?2.5 and θ = 1.5 is displayed in Fig. 5 where the line L4 gives the boundary associated with the last type of instability. Finally, we comment on the e?ects of weak di?usion represented by the term (δ1 +iδ2 )?2 A in Eq. (30). For simplicity, we assume that δ2 is vanishing while δ1 is a small positive. It is clear that this di?usion term simply gives rise to an additional stabilizing term ?δ1 q 2 to each of the eigenvalues λ± as a function of q. Thus, the ?uctuations with su?ciently short wavelength are always decaying. A particularly interesting result from this fact appears in type U2 eigenvalue spectrum in Fig. 3. Depending on the value of δ1 , the dispersion curve is deformed like some curves shown in Fig. 6. Even the dispersion curves of types U1 and U12 could be deformed into a similar form if some parameter vaules are chosen suitably. It is clear that there is a critical value of δ1 at which ?uctuations with a certain wavenumber qc become unstable. Due to the presence of long-wavelength ?uctuations which are almost neutral in stability, the unstable growth of the mode with qc will generally be unable to 14
lead to a Turing type periodic standing pattern. Instead, from the outset, a group of modes with wavenumbers close to qc will couple nonlinearly with another group of modes of almost vanishing q with almost neutral stability, leading to a peculiar spatio-temporal chaos[7, 12]. The simple evolution equation
2 2 ?t u = ??x [? ? (1 + ?x )2 ]u ? (?x u)2,
called the Nikolaevskii equation, gives qualitatively the same eigenvalue spectrum as those in Fig. 6. Actually, it was argued previously that the Turing pattern in this system existing for small positive ? is always unstable, and as a result the system immediately becomes turbulent characterized by the coexistence of turbulent ?uctuations with vastly di?erent length scales. Possibility of this type of turbulence in reaction-di?usion systems was discussed by Fujisaka and Yamada. In electro-convective systems, similar type of complex behavior was discovered by Kai et al. which they called soft-mode turbulence.
If spatially distributed limit cycle oscillators are di?usively coupled not directly but via a certain di?usive component, the oscillators may be viewed as coupled nonlocally after a mathematical elimination of this di?usive variable. We showed in the present paper that this actually occurs in reaction-di?usion systems and that the nonlocality of this kind persists even close to the Hopf bifurcation point provided the coupling between each local oscillator and the di?usive component is su?ciently weak. Speci?cally, under this condition, the systems is reduced to a comlex Ginzburg-Landau type equation with nonlocal rather than di?usive coupling. Temporal nonlocality or memory e?ects which generally exist before reduction does not appear explicitly in the reduced equation, still they may generally a?ect the functional form of the coupling function. Our results were generalized so as to include the e?ects of direct but weak di?usive coupling among the oscillators. Linear stability analysis of the uniformly oscillating state of the reduced equation was also carried out. Some new types of eigenvalues spectrum were found to arise, and their physical relevance was suggested. How the solution behaves in the nonlinear regime is known only partially. For instance, type U12 spectrum in Fig. 3 was found to lead to turbulence with multi-a?nity. It was 15
also found that even the normal spectrum (type S) can give rise to peculiar spiral waves without phase singularity when the nonlocal coupling becomes weak. Di?erent aspects of the nonlocal CGL including the afore-mentioned type of turbulence associated with the dispersion curve in ?g. 6 will be developed in forthcoming papers.
 Y. Kuramoto, Chemical Oscillation, Waves, and Turbulence (Springer, New York, 1984).  M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65 851 (1993).  T. Bohr et al., Dynamical Systems Approach to Turbulence, Cambridge Univ. Press 1998.  Y. Kuramoto, Prog. Theor. Phys. 94 321 (1995).  Y. Kuramoto and H. Nakao, Physica D, 103 294 (1997), Y. Kuramoto, D. Battogtokh, and H. Nakao, Phys. Rev. Lett. 81 3543 (1998), Y. Kuramoto, H. Nakao, and D. Battogtokh, Physica A 288 244 (2000).  J .F. Heagy, L. M. Pecora, and T. L. Carroll, Phys. Rev. Lett. 74 4185 (1995).  M. I. Tribelsky, Phys. Rev. E 54 4973 (1996); Usp.Fiz.Nauk. 167 167 (1997), M. I. Tribelsky and K. Tsuboi, Phys. Rev. Lett. 76 1631 (1996).  S. Kai, K. Hayashi, and Y. Hidaka, J. Phys. Chem. 100 19007 (1996).  V. N. Nikolaevskii, in Recent Advances in Engineering Sience, ed. by S. L. Koh and C. G. Speciale, Lecture Notes in Engineering No.39 (Springer-Verlag, Berlin, 1989), p.210, V. N. Nikolaevskii, Dokl. Akad. Nauk SSSR 307 570 (1989).  H. Fujisaka and T. Yamada, Prog. Theor. Phys. 106 315 (2001).  Y. Kuramoto and S. Shima, to appear in Prog. Theor. Phys. Suppl. 150 (2003).  Similar complex dynamics arising in our 3-component reaction-di?usion model and also in nonlocal CGL is being studied by D. Tanaka (in preparation).
Figure Captions Fig.1 The sign of u(q) changes with q in three di?erent ways. The ?gure shows how the corresponding domains A, B and C appear in the a1 -a2 plane. u(q) is positive for all q in A, negative above some critical q in B, and negative in a ?nite interval of q in C. The line separating domains A and C is given by a parabola a2 = a2 /4a0 . 1 Fig.2 The sign of v(q) changes with q in four di?erent ways. The ?gure shows how the corresponding domains A′ , B′ , C′ and D′ appear in the b0 -b1 plane. v(q) is positive for all q in A′ , negative above some critical q in B′ , negative for all q in C ′ , and negative below some critical q in D′ . Fig.3 Four types of eigenvalue spectrum for the phase-like ?uctuations about the uniform oscillation which is stable only for type S. The ?gure shows how they appear in the K-Kc1 plane in the vicinity of the origin K = 0. The other parameters are ?xed as c2 = 1.0 and θ = 6.0. In each of the four cases, the eigenvalues of the amplitude-like ?uctuations, which is not shown in the ?gure, form a branch completely separated from the phase branch and remain negative for all q. Fig.4 Figure 3 is extended to the region of larger K. A new type of eigenvalue spectrum appears for large negative K. This type, labeled by U3 is similar to U2 in Fig. 3 except that the amplitude branch merges with the phase branch above a certain wavenumber which is still below the wavenumber at which Reλ changes sign. Thus, the eigenvalues associated with the unstable ?uctuations are complex rather than real, i.e., the instability is oscillatory in nature. Fig.5 Similar to Fig. 4 but for di?erent values of c2 and θ, i.e., c2 = ?2.5 and θ = 1.5. New type of eigenvalue spectrum, denoted by U4 , appears which di?ers from U3 in Fig. 4 only in that the eigenvalues saturate to a negative value as q goes to in?nity. Fig.6 Eigenvalue spectra close to a ?nite-wavenumber instability, which are obtained as a modi?cation of U2 in Fig. 3 by assuming nonvanishing δ1 in the reduced equation (30).
a 2=a2 /4a0 1
S U U
-1 -1 0
-4 -1 0